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The propagation of gravitational waves can reveal fundamental features of the structure of spacetime. For instance, differences in the propagation of gravitational-wave polarizations would be a smoking gun for parity violations in the gravitational sector, as expected from birefringent theories like Chern-Simons gravity. Here we look for evidence of amplitude birefringence in the third catalog of detections by the Laser Interferometer Gravitational Wave Observatory and Virgo through the use of birefringent templates inspired by dynamical Chern-Simons gravity. From $71$ binary-black-hole signals, we obtain the most precise constraints on gravitational-wave amplitude birefringence yet, measuring a birefringent attenuation of $\kappa = -0.019^{+0.038}_{-0.029} \, \mathrm{Gpc}^{-1}$ at $100 \, \mathrm{Hz}$ with $90\%$ credibility, equivalent to a parity-violation energy scale of $M_{\rm PV} \gtrsim 6.8 \times 10^{-21}\, {\rm GeV}$.

We expand the class of groups with relatively geometric actions on CAT(0) cube complexes by proving that it is closed under $C'(\frac16)$--small cancellation free products. We build upon a result of Martin and Steenbock who prove an analogous result in the more specialized setting of groups acting properly and cocompactly on Gromov hyperbolic CAT(0) cube complexes. Our methods make use of the same blown-up complex of groups to construct a candidate collection of walls. However, rather than arguing geometrically, we show relative cubulation by appealing to a boundary separation criterion of Einstein and Groves and proving that wall stabilizers form a sufficiently rich family of full relatively quasi-convex codimension-one subgroups. The additional flexibility of relatively geometric actions has surprising new applications. For example, we prove that $C'(\frac16)$--small cancellation free products of residually finite groups are residually finite., Comment: 35 pages, 3 figures

Bowditch characterized relative hyperbolicity in terms of group actions on fine hyperbolic graphs with finitely many edge orbits and finite edge stabilizers. In this paper, we define generalized fine actions on hyperbolic graphs, in which the peripheral subgroups are allowed to stabilize finite sub-graphs rather than stabilizing a point. Generalized fine actions are useful for studying groups that act relatively geometrically on a CAT(0) cube complex, which were recently defined by the first two authors. Specifically, we show that a group acting relatively geometrically on a CAT(0) cube complex admits a generalized fine action on the one-skeleton of the cube complex. For generalized fine actions, we prove a criterion for relative quasiconvexity as subgroups that cocompactly stabilize quasi-convex sub-graphs, generalizing a result of Martinez-Pedroza and Wise in the setting of fine hyperbolic graphs. As an application, we obtain a characterization of boundary separation in generalized fine graphs and use it to prove that Bowditch boundary points in relatively geometric actions are always separated by a hyperplane stabilizer.

We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the classes of groups known to have locally uniform exponential growth to include extensions of either word hyperbolic or right-angled Artin groups by groups with locally uniform exponential growth. From this, we deduce that the automorphism group of a torsion-free one-ended hyperbolic group has locally uniform exponential growth. Our methods also demonstrate that automorphism groups of torsion-free one-ended toral relatively hyperbolic groups and certain right-angled Artin groups satisfy our quantitative subgroup alternative., Comment: Comments are welcome!

We study uniform exponential growth of groups acting on CAT(0) cube complexes. We show that groups acting without global fixed points on CAT(0) square complexes either have uniform exponential growth or stabilize a Euclidean subcomplex. This generalizes the work of Kar and Sageev that considers free actions. Our result lets us show uniform exponential growth for certain groups that act improperly on CAT(0) square complexes, namely, finitely generated subgroups of the Higman group and triangle-free Artin groups. We also obtain that non-virtually abelian groups acting freely on CAT(0) cube complexes of any dimension with isolated flats that admit a geometric group action have uniform exponential growth., Comment: Minor changes to address referee comments. Final version to appear in Algebraic & Geometric Topology

Following the rationale of end-to-end modeling, CTC, RNN-T or encoder-decoder-attention models for automatic speech recognition (ASR) use graphemes or grapheme-based subword units based on e.g. byte-pair encoding (BPE). The mapping from pronunciation to spelling is learned completely from data. In contrast to this, classical approaches to ASR employ secondary knowledge sources in the form of phoneme lists to define phonetic output labels and pronunciation lexica. In this work, we do a systematic comparison between grapheme- and phoneme-based output labels for an encoder-decoder-attention ASR model. We investigate the use of single phonemes as well as BPE-based phoneme groups as output labels of our model. To preserve a simplified and efficient decoder design, we also extend the phoneme set by auxiliary units to be able to distinguish homophones. Experiments performed on the Switchboard 300h and LibriSpeech benchmarks show that phoneme-based modeling is competitive to grapheme-based encoder-decoder-attention modeling., Comment: 5 pages, 6 tables

We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential growth. In addition, we provide a quasi-isometric characterization of hierarchically hyperbolic groups without uniform exponential growth. To achieve this, we gain new insights on the structure of certain classes of hierarchically hyperbolic groups. Our methods give a new unified proof of uniform exponential growth for several examples of groups with notions of non-positive curvature. In particular, we obtain the first proof of uniform exponential growth for certain groups that act geometrically on CAT(0) cubical spaces of dimension 3 or more. Under additional hypotheses, we show that a quantitative Tits alternative holds for hierarchically hyperbolic groups., Comment: Includes appendix by Radhika Gupta and Harry Petyt and improved exposition